Problem 121 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It Does \(y_{1}=\ln [x(x-2)]\) have the same domain as \(y_{2}=\ln x+\ln (x-2) ?\) Explain.

Step-by-Step Solution

Answer

No, \(y_{1}=\ln [x(x-2)]\) and \(y_{2}=\ln x + \ln (x-2)\) do not have the same domain.

Step 1: Identify the Domain for Both Functions

In order for a logarithmic function logarithm to be defined, the argument of the logarithm needs to be greater than zero. For \(y_{1} = \ln [x(x-2)]\), the values for \(x\) must make \(x(x-2) > 0\). This results in two intervals for x: (-∞,0) and (2,∞). For \(y_{2} = \ln x + \ln (x-2)\), each term under the logarithm must be also greater than zero. So, \(x > 0\) for \(\ln x\) and \(x > 2\) for \(\ln (x-2)\). The common domain that satisfies both conditions is (2,∞).

Step 2: Compare the Domains

The domain for \(y_{1}\) is (-∞,0) and (2,∞), and the domain for \(y_{2}\) is (2,∞). The domain for \(y_{2}\) falls within the domain of \(y_{1}\), but \(y_{1}\) has additional values for x (<0) that \(y_{2}\) cannot accept.

Step 3: Conclude

Therefore, \(y_{1}\) and \(y_{2}\) do not have the same domain

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of mathematics, especially in calculus and exponential growth and decay problems. Understanding how these functions work is crucial. A logarithm is the inverse operation to exponentiation, which means it helps us find the exponent as a base ten or often base e (natural logarithm) which gives the result. For the natural logarithm, denoted as \(\ln(x)\), the base is the number \(e\), which is approximately equal to 2.718.When dealing with logarithmic expressions, it's important to note that the argument — the part inside the log — must always be greater than zero. This is because the logarithm of zero or a negative number is undefined in the real number system. When you see something like \(y = \ln[f(x)]\), it means we're interested in finding the values of \(x\) for which \(f(x) > 0\).Logarithms have useful properties that simplify expressions:

• Product Rule: \( \ln(a\times b) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power Rule: \( \ln(a^b) = b \ln(a) \)

Understanding these properties helps in solving logarithmic equations and comparing functions as we see in the exercise.

Interval Notation

Interval notation is a way of representing a set of numbers on the number line. It's quite handy when we need to describe the domain of functions succinctly. The domain of a function is the set of all possible input values (usually \(x\)) for which the function is defined.When using interval notation, there are a few key types of intervals:

• Open intervals: Denoted by parentheses, such as \((a, b)\), which includes all numbers between \(a\) and \(b\), but not \(a\) and \(b\) themselves.
• Closed intervals: Denoted by brackets, such as \([a, b]\), which includes all numbers between \(a\) and \(b\), including \(a\) and \(b\) themselves.
• Half-open intervals: A combination, like \((a, b]\) or \([a, b)\), which includes all numbers between \(a\) and \(b\) but includes only one of the endpoints.
• Infinite intervals: Used when the interval goes on indefinitely. For example, \((-fty, a)\) or \((b, fty)\).

In our exercise, the domains are expressed using interval notation. By determining which intervals are applicable based on the logarithmic function rules, mathematicians can easily communicate the range of valid inputs.

Function Comparison

In mathematics, comparing functions involves analyzing various properties such as domain, range, and behavior. In our exercise, we compare the domains of two logarithmic functions, \(y_{1} = \ln[x(x-2)]\) and \(y_{2} = \ln x + \ln(x-2)\).Here's how the comparison works step by step:

• For \(y_{1}\), the domain is determined by solving the inequality \(x(x-2) > 0\). This results in two intervals where the function is defined: \((-fty,0)\) and \((2,fty)\).
• For \(y_{2}\), each term \(\ln x\) and \(\ln(x-2)\) contributes its constraints: \(x > 0\) and \(x > 2\). The common valid interval is \((2,fty)\).
• Matching these intervals, \(y_{2}'s\) domain \((2,fty)\) is contained fully within \(y_{1}'s\) domain, but \(y_{1} \) includes additional \((-\infty, 0)\).

With this step-by-step comparison, you can see that \(y_{1}\) and \(y_{2}\) do not share the same domain, even though they overlap in \((2, fty)\). This highlights the importance of careful function comparison for accurately determining domains and understanding their implications.